Joyal's CatLab
Natural transformations

**Basic Category Theory** ## Contents * Categories * Functors * Natural transformations * Functors of two variables * Adjoint Functors and Monads

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**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

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Contents

Definition

for every morphism f:ABf:A \to B in C\mathbf{C}.

We shall often write α:FG:CD\alpha:F\to G: \mathbf{C}\to \mathbf{D} to indicate that α\alpha is a natural transformation between two functors CD\mathbf{C}\to \mathbf{D}.

The composite of two natural transformations α:FG\alpha:F\to G and β:GH\beta:G\to H in the category [C,D][\mathbf{C},\mathbf{D}] is the natural transformation βα:FH\beta \alpha:F\to H obtained by putting (βα) A=β Aα A(\beta\alpha)_A=\beta_{A}\alpha_A for every object ACA\in \mathbf{C},

Remark

The category [C,D][\mathbf{C},\mathbf{D}] is locally small if C\mathbf{C} is small and D\mathbf{D} locally small, in which case we shall often denote it by D C\mathbf{D}^{\mathbf{C}}. The category D C\mathbf{D}^{\mathbf{C}} is small, if additionally D\mathbf{D} is small.

Examples

in the category C\mathbf{C}. The category [I,C]=C I[I,\mathbf{C}]= \mathbf{C}^I is called the arrow category of C\mathbf{C}.

Revised on March 29, 2010 at 03:26:10 by joyal